where \(\mathbf {v}^m:=(v_n^m)_{n\in \mathbb {Z}}\)  is given by \(v_m^m=v\)  and \(v_n^m=0\)  for \(n\ne m\) . Then, it is verified in the proof of [1]} that \(h_m\colon \mathbb {R}^d \rightarrow \mathbb {R}^d\)  are \(C^1\) -diffeomorphisms satisfying (REF ).
Moreover, we know that
\(D h_m(v) z=(D\Phi (\mathbf {v}^m)\mathbf {z}^m)_m \quad \text{and} \quad Dh_m^{-1}(v)z=(D\Phi ^{-1}(\mathbf {v}^m)\mathbf {z}^m)_m,\) 
